The vibration study of a sandwich conical shell with a saturated FGP core

This paper is provided to analyze the free vibration of a sandwich truncated conical shell with a saturated functionally graded porous (FGP) core and two same homogenous isotropic face sheets. The mechanical behavior of the saturated FGP is assumed based on Biot’s theory, the shell is modeled via the first-order shear deformation theory (FSDT), and the governing equations and boundary conditions are derived utilizing Hamilton’s principle. Three different porosity distribution patterns are studied including one homogenous uniform distribution pattern and two non-homogenous symmetric ones. The porosity parameters in mentioned distribution patterns are regulated to make them the same in the shell’s mass. The equations of motion are solved exactly in the circumferential direction via proper sinusoidal and cosinusoidal functions, and a numerical solution is provided in the meridional direction utilizing the differential quadrature method (DQM). The precision of the model is approved and the influences of several parameters such as circumferential wave number, the thickness of the FGP core, porosity parameter, porosity distribution pattern, the compressibility of the pore fluid, and boundary conditions on the shell’s natural frequencies are investigated. It is shown that the highest natural frequencies usually can be achieved when the larger pores are located close to the shell’s middle surface and in each vibrational mode, there is a special value of the porosity parameter which leads to the lowest natural frequencies. It is deduced that in most cases, natural frequencies decrease by increasing the thickness of the FGP core. In addition, reducing the compressibility of the porefluid a small growth in the natural frequencies can be seen.

Due to the numerous use of the conical shells in different engineering applications such as aerospace and mechanical engineering, high-power aircraft jet engines, high-speed centrifugal separators, and gas turbines, a considerable number of investigations have been presented on the mechanical analysis of such structures, recently. Sofiyev 1 investigated the stability and free vibration analyses of heterogenous composite truncated conical shells reinforced with carbon nanotubes (CNTs) subjected to an axial load. He examined the effects of the percentage of the CNTs and heterogeneity on the buckling and free vibrational characteristics of the shell. The free vibration characteristics of rotating polymeric truncated conical shells enriched by graphene nanoplatelets (GNPs) were investigated by Afshari 2 . It was shown by him that the sequence of vibrational modes can be affected by the variation of semi-vertex angle. By utilizing analytical and numerical techniques and experimental tests, the free vibration study of conical shells stiffened by bevel stiffeners was examined by Zarei et al. 3 . They studied the influences of the shell's geometrical characteristics on the natural frequencies of such a structure. Yousefi et al. 4,5 studied the forced and free vibrational behavior of three-phase CNT/polymer/fiber truncated conical panels and shells. It was revealed by them that the larger length and higher embracing and semi-vertex angles result in the smaller natural frequencies. To complete these works, they hired particle swarm optimization to find the best values of mass fractions of the CNTs and fibers and orientation of the fibers to minimize the cost and maximize the fundamental frequency of the three-phase CNT/polymer/fiber laminated truncated conical panels 6 . Aris and Ahmadi 7 studied the analysis of the nonlinear resonance of FGM (functionally graded materials) truncated conical shells exposed to an external harmonic excitation and thermal loading. They examined the effects of the shell's geometrical characteristics and temperature on the nonlinear vibrational characteristics of the shell. By incorporating the agglomeration of the CNTs, the free vibration study of a CNT-reinforced spinning truncated conical shell was examined by Afshari and Amirabadi 8 . It was shown by them that variation of the rotational speed may change the sequence of the vibrational modes. The vibration study of combined conical-ribbed cylindrical-conical shell structures was investigated by Zhang et al. 9 . They approved the precision of their work www.nature.com/scientificreports/ Material properties. As depicted in Fig. 2, three different porosity distribution patterns are considered in this paper for the shell's core including a uniform porosity distribution (UD) and two non-uniform ones (SI and SII). The elastic modulus of the FGP core varies along thickness direction as 36 where E 0 is the modulus of elasticity of the material with no porosity and e 0 , e 1 , and e 2 stand for the porosity parameters.
To have a fair comparison between these porosity distribution patterns, it is better to regulate the porosity parameters to create the same value of mass. Utilizing the following equation between density (ρ) and the elastic modulus 37 in which ρ 0 represents the density of the material with no porosity, one can write   For some values of the porosity parameter e 1 , the corresponding values of the porosity parameters e 0 and e 2 can be found in Table 1 and can be approximately stated as bellow 36 : Based on Biot's poroelasticity theory for isotropic poroelastic mediums, the constitutive relations are stated as 16 in which σ ij and ε ij sequentially stand for the stress and strain tensors and α 0 is the Biot coefficient of effective stress. Also, with the following definitions, G, λ u , p, ε kk , and δ ij represent the shear modulus, undrained Lame parameter, pore fluid pressure, volumetric strain, and the Kronecker delta: in which ξ stands for the variation of fluid volume content and ν u and M sequentially represent undrained Poisson's ratio and Biot's modulus and are defined as below: where B 0 is known as the Skempton coefficient which deputizes the compressibility of the pore fluid. By increasing the Skempton coefficient from zero to one, the pore fluid varies from a completely compressible fluid to a noncompressible fluid.
Governing equations. As stated in the FSDT, the following relation can be utilized for the displacement field 38 : in which u 1 , u 2 , and u 3 sequentially are the displacement components along of x, θ, and z, directions; u, v, and w show the corresponding components of displacement at the middle surface (z = 0); and φ x and φ θ stand for the rotations about θ-and x-axes, sequentially. (4) e 0 = 1.944e 6 1 − 3.417e 5 1 + 2.278e 4 1 − 0.6708e 3 1 + 0.122e 2 1 + 0.6362e 1 , e 2 = − 0.4269e 3 1 − 0.009286e 2 1 + 1.732e 1 . www.nature.com/scientificreports/ The non-zero components of the strain tensor (ε ij ) can be presented as below 38 : The governing equations and boundary conditions can be derived utilizing Hamilton's principle as below 39 : in which [t 1 ,t 2 ] is an arbitrary time interval, δ shows the variational operator, T shows kinetic energy, U s is strain energy, and W n.c. stands for the work done by external non-conservative loads. The shell's kinetic energy can be presented as below: where in which dS = rdxdθ shows the shell's surface. Equation (14) can be represented using Eqs. (11) and (15) as below: where Due to the symmetry in pore distribution patterns, it is evident that I 1 = 0, consequently, the variation of the shell's kinetic energy can be stated as below: The variation of the shell's strain energy can be calculated as below which can be represented using Eqs. (12) and (15) as where By substituting Eqs. (8) and (12) into Eq. (21), the following equation can be achieved: (12) ε xx = ∂u ∂x + z ∂ϕ x ∂x ,ε θθ = sin α r u + cos α r w + 1 r ∂v ∂θ + z sin α r ϕ x + 1 r ∂ϕ θ ∂θ , Due to the symmetry in pore distribution patterns, it is evident that B ij = 0, consequently, Eq. (22) can be simplified as follows: In the free vibration analysis, the shell is not subjected to any external load (δW n.c. = 0); consequently, by substituting Eqs. (18) and (20) into Eq. (13), the following set of the governing equations can be achieved: and the corresponding boundary condition can be stated as below: By substituting Eq. (24) into Eq. (25) and employing the following solution: in which ω is an eigenvalue and n is called the circumferential wave number, the set of the governing equations can be represented as below:  (24) and (27) into Eq. (26), the boundary conditions can be stated as below:

Solution methodology
In the current section, the DQM is employed as a well-accepted and well-known numerical approach to provide an approximate solution for the set of the governing Eqs. (28) with any combination of the boundary conditions (29) at both ends of the shell (x = 0,L). Based on the main idea in the DQM, each derivative of a function like P(x) can be estimated in terms of the weighted sum of its values at a set of grid points as below: where [F (k) ] is called the weighting coefficient matrix related to the kth order derivative and is presented as below 40 : The distribution pattern of the grid points plays an important role in the convergence of the solution in the DQM. With the following definition for 0 ≤ x ≤ L, the Gauss-Lobatto-Chebyshev distribution pattern is utilized in this work 40 : By applying Eq. (30) and the following notation: where where Γ 11 − Γ 55 are associated with the condition at x = 0 and Γ 61 − Γ 105 are associated with the condition at x = L. For a truncated conical shell clamped at the small radius (x = 0) and simply supported at the large radius (x = L) which is denoted by "CS" in this paper, Γ 11 − Γ 105 are presented as below: where subscripts 1 and N respectively stand for the first and last rows of each matrix.
Simultaneous solutions of Eqs. (34) and (37) generate an inequality between the numbers of the equations and unknown variables (non-square matrices in the final eigenvalue equation) 38 . To remove this inequality, let us divide the grid points into two sets: the boundary points (x 1 and x N ) and the domain ones (x 2 , x 3 ,…, x N−2 , x N−1 ). Ignoring the satisfaction of governing equations at the boundary points, Eq. (34) can be represented as below: in which the sign ~ is utilized to show the created non-square matrices.
Obviously, ignoring the satisfaction of governing equations at the boundary points decreases the accuracy of the solution. But, in Gauss-Lobatto-Chebyshev distribution pattern (Eq. (32)), there is an agglomeration of the points at two ends of the domain which contains the boundary points. Consequently, the side effect of the above-mentioned assumption dramatically decreases 38,41 .
By partitioning the matrices to separate the columns associated with the boundary and domain points, Eqs. (37) and (40) can be represented as below:    27)), and the second one (m) is employed to indicate the meridional mode number. Also, the following definition is utilized in this paper to present the natural frequencies in a dimensionless form: where ρ f and E f sequentially stand for the density and elastic modulus of the face sheets.

Numerical results
Numerical results are provided in this part of the paper for the presented solution. In what follows, except as expressly stated, a CS conical shell is considered with the geometrical characteristics a = 0.5 m, α = 45°, h/a = 0.1, L/a = 4, and h c /h = 0.5. The shell consists of an FGP core of distribution pattern SI, e 1 = 0.5, and B 0 = 0.5. The mechanical properties of the FGP core are ρ 0 = 2700 kg/m 3 , ν = 0.25, E 0 = 60 GPa, and α 0 = 0.19 16,42 and those of the face sheets are ρ f =2707 kg/m 3 , ν f =0. 3, and E f =70 GPa.
Convergence analysis and validation. The convergence analysis of the presented solution is examined in Fig. 3 for some vibrational modes. This figure shows that as the number of grid points grows (N in Eqs.  Table 2 versus those reported by Liew et al. 43 . As this table reveals, the results are in high matching which confirms the precision of the presented solution.  Table 3 against those reported by Dai et al. 44 . This table confirms that results are in high matching which confirms the precision of the presented numerical solution. Parametric study. The dependency of the shell's natural frequencies on the circumferential wave number is examined in Fig. 4. As this figure reveals, by increasing the circumferential wave number, the natural frequencies experience an initial reduction followed by increasing growth. In other words, there is a special value of the circumferential wave number which provides the lowest natural frequency (the fundamental frequency, λ n1 ). As the circumferential wave number increase, the shell experiences various shapes of harmonic functions (sinus Table 2.  www.nature.com/scientificreports/ or cosinus) in the circumferential direction. Depending on the geometrical parameters of the shell, boundary conditions, and the meridional mode number (m), there is a specific shape of the shell in the circumferential associated with a specific value of the circumferential wave number which provides the minimum rigidity and consequently the minimum natural frequency. Table 4 shows the influences of the boundary conditions on the shell's natural frequencies. As observed, the highest natural frequency at each vibrational mode belongs to the CC shell which means that the more constrained boundaries result in higher natural frequencies. It can be explained by the higher rigidity of the shell under CC boundary conditions. A comparison between the CS, SC, and FC shells reveals that the natural frequencies of the SC shell are greater than the corresponding ones of the CS shell and in some vibrational modes, the natural frequency of the FC shell is greater than the natural frequency of the CS one. It means that the boundary condition at x = L (the shell's large radius) has a stronger effect on the natural frequencies of the conical shells rather than the boundary condition at x = 0 (the shell's small radius). It can be explained by the higher perimeter of the shell's edge at its large radius in comparison with the small one. Table 5 is presented to investigate the effect of the pore distribution pattern on the shell's natural frequencies. As this table shows, in most vibrational modes, the highest natural frequency belongs to the SI pattern. As depicted in Fig. 2, in the SI pattern the big pores are distributed close to the shell's neutral surface (SI) which leads to the minimum reduction in the shell's flexural rigidity. It is noteworthy that alongside the flexural rigidity, the rotational inertia (I 2 in Eq. (17)) can be affected by the pore distribution pattern. Consequently, in some cases, the highest natural frequency belongs to the SII pattern which has the minimum rotational inertia.
The dependency of the shell's natural frequencies on the porosity parameter is examined in Fig. 5. By growing the porosity parameter, the size of the pore increases which decreases both the rigidity and inertia of the shell. Consequently, as the porosity parameter grows, depending on the vibrational mode, either increase or decrease in the natural frequency can be seen. As shown in this figure, due to the confrontation between the reductions in the rigidity and inertia of the shell, an increase in the porosity parameter has no remarkable effect on the natural frequencies. Thus, to make it possible to show the small variations of the natural frequencies in different vibrational modes, simultaneously, the following frequency parameter is defined:  Table 5. Dependency of the natural frequencies on the pore distribution pattern. Significance values are given in bold. www.nature.com/scientificreports/ Figure 5 confirms that for (n,m) = 1, 2, 3, 4, by increasing the porosity parameter from zero to e 1 = 0.6, the maximum reduction and increase in the natural frequencies are less than 5% and 1.5%, respectively.
For a specified value of the shell's thickness, Fig. 6 shows the effect of the thickness of the FGP core on the shell's natural frequencies. By increasing the thickness of the FGP core, both inertia and rigidity of the shell decrease. Thus, as the thickness of the FGP core grows, depending on the vibrational mode, either increase or decrease in the natural frequency can be seen.
As this figure shows, due to the confrontation between the reductions in the inertia and rigidity of the shell, an increase in the shell's thickness has no considerable influence on the natural frequencies. Consequently, to make it possible to show the small variations of the natural frequencies in various vibrational modes, simultaneously, the frequency parameter is defined as below: www.nature.com/scientificreports/ Figure 6 shows that for (n,m) = 1,2,3,4, by increasing the thickness of the FGP core from zero to 0.8 h, the maximum reduction in the natural frequencies is less than 12%. Figure 7 is provided to examine the effect of the compressibility of the pore fluid on the shell's natural frequencies. As observed, by increasing the Skempton parameter (decreasing the compressibility of the pore fluid), a small growth occurs in the natural frequencies which can be explained by the small growth in the rigidity of the shell. As this figure shows, for (n,m) = 1, 2, 3, 4, by increasing the Skempton parameter from the minimum possible value (B 0 = 0) to the maximum possible value (B 0 = 1), the maximum increase in the natural frequencies is less than 1%.
It should be noticed that due to the weak effect of the compressibility of the pore fluid on the natural frequencies, to make it possible to show the small variations of the natural frequencies in different vibrational modes, simultaneously, the frequency parameter is defined as follows:

Conclusions
The free vibrational analysis of a sandwich truncated conical shell with a saturated FGP core and two same homogenous isotropic face sheets was examined. The mechanical behavior of the saturated FGP core and mathematical modeling of the shell was performed based on Biot's theory and the FSDT respectively. Three different distribution patterns of the pores were investigated including a uniform distribution pattern and two nonhomogenous symmetric ones. The main findings of the paper can be listed as below: • The more constrained boundaries at the ends of the shell result in higher natural frequencies.
• The boundary condition at the shell's large radius has a stronger effect on the natural frequencies of the conical shells rather than the boundary condition at the shell's small radius. • When the bigger pores are located close to the neutral surface of the shell, the natural frequencies become greater. • By increasing the porosity parameter and thickness of the FGP core, either growth or reduction in the natural frequency can be seen. It depends on the vibrational mode. • The lower compressibility of the pore fluid results in higher natural frequencies. But, the maximum increase is less than 1%.  Figure 7. Dependency of the natural frequencies on the compressibility of the pore fluid.